State phase, rules, and theorems

The interface response theory enables us to find all the states of a final system once one knows those of any corresponding complete reference system. Its state phase is the essential element enabling us to achieve these results. It checks the state number conservation between the reference and final systems. It also provides the eigenvalues of the shifted states. The final system response function provides the corresponding eigenfunctions, their continuity rules, and their robust nodes. These robust zeros enable us to derive bound in continuum (BIC), semi-infinite bound in continuum (SIBIC), long-lived resonance, and state activation theorems